Maximum Entropy of Sums of Independent Ternary Random Variables

The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet is considered and settled in the ternary case. Namely, the following theorem is established: if $X_1,\ldots,X_n$ are independent random variables taking values in $\{0,1,2\}$, then the entropy of $S_n=X_1+\cdots+X_n$ is maximized when $X_1,\ldots,X_{n-1}$ are uniform on $\{0,2\}$ and the probability mass function of $X_n$ is given by $\Prob(X_n=0) = \Prob(X_n=2) = w/2$, $\Prob(X_n=1) = 1-w$, where $w = \big(1 + 2^{-H(B_n)+H(B_{n-1})}\big)^{-1}$ and $B_m\sim \Bin(m,1/2)$. The statement can be seen as an extension to ternary alphabets of the Shepp--Olkin--Mateev theorem. The proof uses the Hermite--Biehler theorem, Newton's inequalities, and Yu's maximum-entropy theorem for ultra-log-concave distributions.